Energy levels in rare-earth ions

The tensor-operator and group-theoretical methods of Racah are used to examine the following contributions to the Hamiltonian of a rare-earth ion: (a) the Coulomb interaction between the electrons, (b) the spin-orbit coupling of the electrons, and (c) the term arising from the influence of an external electrostatic field such as occurs when the rare-earth ion is situated in a crystal lattice. With regard to (a), the positions of all the terms of the configurations f$^{2}$, f$^{3}$ and f$^{4}$, as well as all the terms of other configurations f$^{n}$ whose multiplicities are the highest or the highest but one, are tabulated on the basis of a hydrogenic 4f wave function. The theory of (b) is developed, and the parameter $\lambda $ defining the spin-orbit splitting of a term is given for all terms whose energies have been found. (c) is treated by similar techniques and the theory is related to the one at present in use. All the general methods are illustrated by examples.